We begin with a brief review of the mathematical description of decoherence in quantum systems, and refer to some of the well-known textbooks [1]}, [2]} for further details. Noise in quantum systems is described by the mathematical framework of quantum operations, often referred to as quantum channels in the context of quantum computation [1]}. A quantum channel is a completely positive trace-preserving (CPTP) map describing the non-unitary evolution of an open quantum system interacting with its environment, as shown in Fig. REF . The action of such a map \(\mathcal {E}\)  on the state \(\rho \)  of a quantum system can be described via a set of Kraus operators \(\lbrace E_{i}\rbrace \) , as,
\(\mathcal {E}(\rho ) = \sum _{i}E_{i}\rho E_{i}^{\dagger }\) , satisfying the normalization condition, \(sum_i E_i^\dagger E_i = I\) . These Kraus operators essentially represent the different error operators associated with the noise channel \(\mathcal {E}\) .
